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Recall that given a commutative ring $R$ and a subset $S$ that is multiplicatively closed and does not contain any zero divisors, we can then form the ring of fractions $S^{-1}R$ by formally inverting elements of $S$ . $S^{-1}R$ has the following universal property: there is a ring homomorphism $\phi: R\to S^{-1}R$ with the property that
$\phi(s)$ is a unit in $S^{-1}R$ for every $s\in S$ ;
and if $\sigma: R\to T$ is another ring homomorphism with the above property, then we get a unique ring homomorphism $\delta: S^{-1}R\to T$ such that $\delta\circ \phi=\sigma$ . The category of fractions is the generalization of this concept to category theory.
Definition. Let $\mathcal{C}$ be a category, and $\Sigma$ a class of morphisms in $\mathcal{C}$ . A category of fractions of $\mathcal{C}$ over $\Sigma$ is a pair $(\mathcal{D},F)$ , where
- $\mathcal{D}$ is a category and $F:\mathcal{C} \to \mathcal{D}$ is a functor, such that
- if $(\mathcal{E},G)$ is another such a pair satisfying condition 1 above, then there is a unique functor $H:\mathcal{D} \to \mathcal{E}$ with $H\circ F = G$ .
Equivalently, consider the (large) category $\mathcal{Q}$ with objects all pairs $(\mathcal{D},F)$ satisfying condition 1 above, and a morphism from $(\mathcal{D}_1,F_1)$ to $(\mathcal{D}_2,F_2)$ is a functor $G:\mathcal{D}_1\to \mathcal{D}_2$ where
is a commutative diagram. An initial object in $\mathcal{Q}$ is called a category of fractions (of $\mathcal{C}$ over $\Sigma$ ).
It is clear that a category of fractions is unique up to natural isomorphism, so we call $(\mathcal{D},F)$ the category of fractions of $\mathcal{C}$ over $\Sigma$ , and we denote the category $\mathcal{D}$ by $\mathcal{C}\Sigma^{-1}$ .
For example, let $\mathcal{C}$ is the category with objects $A,B$ and morphisms $1_A,1_B$ and $f:A\to B$ , and $\Sigma=\lbrace f\rbrace$ . Consider the category $\mathcal{D}$ with the same two objects $A,B$ and morphisms $1_A,1_B, g:A\to B$ and its inverse $g^{-1}:B\to A$ , and the functor $F:\mathcal{C}\to \mathcal{D}$ given by $F(A)=A, F(B)=B$ and $F(1_A)=1_A,F(1_B)=1_B$ and $F(f)=g$ . Then $(\mathcal{D},F)$ is the category of fractions of $\mathcal{C}$ over $\Sigma$
. To see this, suppose $G:\mathcal{C}\to \mathcal{E}$ is another functor with $G(f)$ an isomorphism. Define $H:\mathcal{D}\to \mathcal{E}$ given by $H(A):=G(A),H(B):=G(A)$ and $H(1_A)=G(1_A), H(1_B)=G(1_B), H(g)=G(f)$ , and $H(g^{-1})= G(f)^{-1}$ . It is clear that $H$ is a functor with $H\circ F=G$ , and that $H$ is uniquely determined.
In fact, one can prove the following existence property:
Proposition 1 $\mathcal{C}\Sigma^{-1}$ exists if $\Sigma$ is a set. Furthermore, $\mathcal{C}\Sigma^{-1}$ is small if $\mathcal{C}$ is.
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![$\displaystyle \xymatrix@+=1.5cm{& \mathcal{C} \ar[dr]^{F_1} \ar[dl]_{F_2} & \\ \mathcal{D}_1 \ar[rr]_{G} & & \mathcal{D}_2 }$](http://images.planetmath.org:8080/cache/objects/11201/js/img1.png "$\displaystyle \xymatrix@+=1.5cm{& \mathcal{C} \ar[dr]^{F_1} \ar[dl]_{F_2} & \\ \mathcal{D}_1 \ar[rr]_{G} & & \mathcal{D}_2 }$")